Logarithmic Solutions Research Articles

An Analytical Study of Pseudotime for Pressure Drawdown in a Gas Reservoir

Summary The pseudotime transformation in the diffusion equation for pseudopressure is studied analytically for drawdown in a gas reservoir. The equation is not effectively linearized for early drawdown in the infinite-acting period because correction terms in the transformed equation cannot generally be neglected compared with terms interpreted as giving rise to liquid-type behavior. The remaining possibility of approximately logarithmic solutions in pseudotime is considered with slope differing from the liquid-type value. Analytical estimates for the magnitude of correction terms and for deviation of approximate slope are given.

Logarithmic solutions of linear differential systems with a singularity of the first kind† ‡

Logarithmic solutions of linear differential systems with a singularity of the first kind† ‡

Lie Theory and Some Special Solutions of the Hypergeometric Equations

To obtain all solutions of the differential equations of hypergeometric type as basis functions corresponding to models of Lie algebra representations it is necessary to consider certain reducible representations. These representations are classified and models are constructed. A number of special hypergeometric functions arise naturally in the analysis, including the error and incomplete gamma functions, the incomplete beta functions, Legendre functions of the second kind, and some logarithmic solutions of the hypergeometric equation.

Logarithmic solutions of higher-order hypergeometric differential equations

Logarithmic solutions of higher-order hypergeometric differential equations

On logarithmic particular solutions of the equation of cyclic strain of a shallow conical shell

On logarithmic particular solutions of the equation of cyclic strain of a shallow conical shell

On the Solution of Linear Homogeneous Differential Equations

A standard way of solving linear homogeneous differential equations with polynomial coefficients is the series method of Frobenius. The differential equations of the generalized hypergeometric functions, when solved by this method, have non logarithmic series solutions whose coefficients are found directly by solving a two-term recurrence relation as well as logarithmic solutions that are derivable from a two-term recurrence relation. We are here interested in a converse of this statement: THEOREM: If the non-logarithmic solutions of a linear homogeneous differential equation with polynomial coefficients are series whose coefficients are solutions of a two-term recurrence relation, then the differential equation is essentially a generalized hypergeometric differential equation and the solutions are generalized hypergeometric series or logarithmic solutions associated with generalized hypergeometric serses. This theorem is well-known to workers in the special functions of analysis but since it has seemed to be such a surprising result to others, we believe that this theorem should be made a matter of public record. Recurrence relations with more than two terms are so difficult to solve explicitly that virtually all the examples of the Frobenius method given in the standard textbooks that yield series with explicitly given coefficients are examples covered by this theorem. Our analysis will rest heavily on the use of the operator O=x(d/dx). It is a linear operator having the properties Oxn = nXn and 0 [on ]f(X) =-on+lf(X). Let { Pk(6) }I = be a set of polynomials in 0 with coefficients independent of x and such that the polynomials of greatest degree have degree precisely t. Then the most general linear homogeneous differential equation with polynomial coefficients of degree t can be written

XIX. The logarithmic solutions of linear differential equations

XIX. The logarithmic solutions of linear differential equations

On the logarithmic solutions of the generalized hypergeometric equation when 𝑝=𝑞+1

where 6 = z{d/dz) and where the at and ct are any constants, real or complex, the only restriction being that one of the ct must be equal to unity. Such solutions can be found in a number of places in the literature, f But in attempting to study the logarithmic cases of the problem treated in the above-mentioned paper, the author was unable to find the logarithmic solutions of equation (1) in the literature. I t is the purpose of this paper to present these logarithmic solutions, but for the sake of completeness, the non-logarithmic solutions are also given. The methods used are those of Frobenius.J

On Regular Logarithmic Solutions of Linear Differential Equations

On Regular Logarithmic Solutions of Linear Differential Equations

Logarithmic Solutions of Bianchi's Equation

The partial differential equation ∂^nV/∂x1∂x2…∂xn = MV was discussed by Bianchi(1) with the aid of the methods of Riemann and Picard. The results were extended to a more general equation which was also studied by Niccoletti.(2) The original equation, for a constant value of M, was studied later by Sibirani(3) in connection with a generalization of the Bessel function and some partial differential equations were listed which could be solved with the aid of this function. The case in which M is constant has also been studied by Chaundy(4) who gives some solutions in the form of definite integrals which we wish to obtain here with the aid of Murphy's theorem.

Societies and Academies

PARIS Academy of Sciences, Sept. 8.—W. Vernadsky: The radium in aquatic organisms. The amounts of radium in different species of Lemna growing in lakes containing known amounts of radium have been determined. The concentration of radium in the living plant is 100-650 times that present in the water. The amounts of radium vary considerably with the species of Lemna, and this variation does not depend on the weight of the individual plant. The question as to what is the function of radium in the vital processes of Lemna still remains unanswered.—Auguste Lumiere and Mme. R. H. Grange: The protective action of cholesterol against shock caused by flocculates. The facts cited prove that injections of cholesterol protect the animal (rabbit) against shocks of the anaphylactic type.—Joergen Rybner: Nomograms for transformations between rectangular and polar co-ordinates and for complex hyperbolic functions.—Cl. Chevalley: The theory of normic residues.—Radu Badesco: Logarithmic solutions of an integral equation.—Luca Teodoriu: A partial differential equation which occurs in the problem of average.—F. Charles and J. Flandrin: Contribution to the study of Cretaceous soils in the north of Anatolia (Asia Minor).—P. Fallot, A. Marin, and M. Blumenthal: The limestone chain of the Spanish Rif between Xauen and oued M'ter.—Th. Bieler-Chatelan: The polysynthetic quaternary glacier of Monti Simbruini (Central Apennines). The causes of its extension. The author concludes that in the Apennines, in spite of the altitudes being lower than the Alps, the quaternary glaciers could have reached dimensions comparable with those of Alpine glaciers, this being due to the heavy rainfall which has always characterised these ranges.—Jules Amar: The diaphragm origin of respiration.—Angelo Migliavacca: The lipochrome interstitial cells of the uterus.—Re my Collin and Pierre Florentin: The growth of the nuclei in geometrical progression in LcewenthaPs gland.—Mme. Y. Khouvine, E. Aubel,and L. Chevillard: The mechanism of the transformation of pyruvic acid into lactic acid in the liver.